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Linear Algebra  Basis 
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#1
Oct405, 05:41 PM

P: 188

Hello... im doing this problem with basis. Infact, im having a lot of problems understanding basis, i did every question in the text book and I still get seem to understand the idea of it.
So i was hoping somebody can help me with the whole idea about it. Say, for example, how would i go abouts a question like this: Let F be a field and let V = F^3. Let W = {(a1 a2 a3) E F^3 / 2a1  a2  a3 = 0 } Find a basis for W  If a question like that came on a test, id fail it  sad to say. It would also be good if somebody knows a good website or has sameple tests that covers this mataril so that i may get used to it. Thanks 


#2
Oct405, 05:56 PM

P: 429

so any element in W can be represented like so:
w = (a1, a2, a3), where a1, a2, and a3 are arbitrary. but W has the additional restriction that a1 = 1/2 (a2 + a3). so w= ( 1/2 (a2+a3), a2, a3). w = a2 ( 1/2, 1, 0) + a3 (1/2, 0, 1). (it's easy to see that this is the same as above.) so w = span{(1/2, 1, 0), (1/2, 0, 1)}. and that set {(1/2, 1, 0), (1/2, 0, 1)} is our basis. ah, i miss these problems! 


#3
Oct405, 06:04 PM

Emeritus
Sci Advisor
PF Gold
P: 11,155

Keep in mind that the above is not the only basis.
Also, notice that the given vector space is nothing but a (generalization of a) plane through the origin in [itex]\mathbb{R}^3[/itex]. Any pair of vectors in the plane will serve as a basis. 


#4
Oct405, 09:06 PM

P: 188

Linear Algebra  Basis
Hey.. i was just wondering about Brad Barkers post above...
he said that a1 = 1/2 (a2 + a3). well.. shouldn't it be a1 = 1/2 (a2  a3) ? Does it make a difference? 


#5
Oct505, 01:06 AM

Emeritus
Sci Advisor
PF Gold
P: 11,155

No, you said "2a1  a2 a3 = 0"
That gives 2a1 = a2 + a3 


#6
Oct505, 04:49 AM

P: 188

oh that was my silly mistake.. but either way.. i still learned something :)



#7
Oct505, 11:16 AM

P: 188

Hey.... what about the subspace...
U = (a+b+c=0/ a b c is in the Real Numbers) How would you show the span of that. Also, the comment Gokul43201 made, about the basis being the plane through the origin, how did he know that? I mean its a plane because of the two vectors, but how did he know that its through the origin? 


#8
Oct505, 02:54 PM

P: 131

[itex]D = 0 \Longleftrightarrow[/itex] the plane goes through the origin. 


#9
Oct505, 05:36 PM

P: 188

"and that set {(1/2, 1, 0), (1/2, 0, 1)} is our basis." Can that be a Basis for F^3? what is F (Feild)? isnt that the same as R, like R^3



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